Addition & Subtraction — Counting On, and Completing the Ten
Put out seven pebbles. Then put out five more. Now count all of them, one at a time, from the start: one, two, three... twelve. That works. It also gets slow fast — try it with 47 pebbles and 38 more, and you'll be counting for a while. Addition is a way to skip the counting once you already know how many you started with.
See it first
Here's what "add 5" looks like as a picture, not just a sentence — one hop from 7, landing on 12.
7 + 5 is one jump of 5, landing on 12 — not a new fact to memorize, a place to land.
That's the whole idea of addition: a jump of a fixed size. 7 + 5 isn't a
fact you memorize out of nowhere — it's "start at 7, jump 5, see where you
land." Subtraction is the same jump, run backward: 12 - 5 is "start at 12,
jump 5 the other way."
❓ Cross-question — "Isn't that just what addition already means?" Yes — and that's the point of drawing it. Once addition is a jump you can picture, two things most people learn as separate rules turn out to be one idea seen from two sides: subtraction is just addition's undo.
Counting on, and why it stops being enough
For small numbers, "counting on" — starting at 7 and counting five more: 8, 9,
10, 11, 12 — works fine. It's also exactly as slow for 47 + 38 as it is fast
for 7 + 5: forty pebbles is still forty steps, whether you're paying
attention or not. Counting on doesn't get worse as the numbers grow — it just
stays the same amount of slow, one count per step, no matter how big the
numbers are. What you need isn't a faster way to count. It's a way to skip most
of the counting.
Completing the ten
Here's the shortcut. Ten is a number you already know how to add onto without
thinking — 10 + 4 is just 14, no counting required. So the trick is: get to
a ten first, then add whatever's left.
Once you know 8 needs 2 to reach 10, split the 6 you're adding into 2 + 4:
8 + 6
= 8 + 2 + 4 (split 6 into the part that completes ten, and the rest)
= 10 + 4 (8 + 2 is exactly ten)
= 14 (and adding onto a ten is easy)Here's that whole journey as one picture — two jumps instead of six single steps:
The first jump completes the ten. The second jump adds whatever's left. Two hops, same answer as counting six times.
The habit is always the same three moves: find the gap to the next ten, split
the second number by that gap, then add onto the ten. It works whichever number
you start from — try it on 9 + 7 in your head before reading on: 9 needs 1 to
reach 10, so split 7 into 1 + 6, and 10 + 6 = 16.
Subtraction, the same way, backward
Subtraction completes the ten too — you just walk down to it instead of up.
13 - 5: first get from 13 down to 10 (that's 3), then take off what's left
of the 5 (that's 2 more):
The first jump gets down to the ten. The second jump takes off what's left of the 5. Same two-hop idea, run backward.
The same two moves work at any size. 72 - 38: first get from 72 down to 70
(that's 2), then take off what's left of the 38 (that's 36 more):
72 needs 2 to reach 70. Take that off the 38 first, then the remaining 36 comes off the ten.
Why this matters more as the numbers grow
For 7 + 5, counting on and completing the ten take about the same effort —
you might not even notice a difference. The gap shows up on bigger numbers.
47 + 38 by counting on is 38 individual steps. By completing the ten, it's
one small subtraction (47 needs 3 to reach 50), one small addition (38 minus
that 3 is 35), and one easy sum (50 + 35 = 85) — three quick moves, however big
the numbers get. The technique doesn't get harder as the numbers grow. Counting
one at a time does.
The first jump completes the ten. The second jump adds what's left of 38. Two hops instead of thirty-eight single counts.
In the wild
This is the same shortcut a cashier uses making change, and the same one behind "round up, then adjust" estimates in everyday arithmetic — round 47 up to 50, do the easy sum, then correct for the rounding. Once completing the ten is automatic, it stops being a trick you perform and becomes the way you already do arithmetic.
🎯 Your turn
Before you go on: pick two two-digit numbers of your own, and add them using completing-the-ten — out loud or on paper, three moves: the gap, the rest, the easy sum. Then explain, in the exercise below, why splitting the second number that particular way is always allowed.